Efficient and Expressive Equivariance Discovery and Enforcement using Vector Fields

Symmetry plays an increasingly important role in the design and analysis of machine learning models, offering improvements in both generalizability and interpretability. Recent work has demonstrated the discovery and enforcement of continuous symmetries using the Lie derivative along vector fields; however, such approaches either do not explore expressive symmetries, are limited to the topic of invariance, or rely on heuristic elbow curve criteria to detect the number of symmetries, leaving nonlinear equivariance and practical scalability underexplored. In this work, we build upon the vector field approach in two significant ways. First, we introduce a PCA-enhanced formulation for discovery that improves numerical stability and computational efficiency, while replacing heuristic elbow curve selection with principled explained variance criteria. Second, we generalize the approach from invariance to equivariance, enabling nonlinear continuous discovery and enforcement in the context of equivariance. Our empirical results demonstrate a distinct computational advantage, being at least an order of magnitude faster than competing methods. We further demonstrate that vector field regularization can improve generalizability under out of sample test distributions, providing a flexible alternative to approaches that enforce equivariance architecturally. Our results thus establish the vector field approach as a scalable, practical, and flexible tool for symmetry-based learning beyond invariance.